Digitizer target threshold levels
In Gianni's
ALMA Memo 583, footnote 2 indicates that the target level for 2-bit TDM mode should be 1.4 dB higher than FDM. The explanation for this is given below in this email of September 12, 2013.
Hi Todd,
I believe you are correct about the reason for the 1.70 value.
In fact, note that on p. 12 of Memo 583, in the paragraph immediately
following Eq. 23, Gianni mentions a "nominal level of \sigma=1.706 times
the threshold spacing". Here he is referring to the same constant as
in Footnote 2, I believe, but with four significant digits rather than
three. To six significant digits, the optimal voltage threshold for
8-(equispaced) level sampling is 0.586019\sigma
(
ALMA memo 591). The reciprocal of
0.586019 is 1.70643. [The 1.706 value is also used on p. 13 of the
memo.]
For 4-level quantization, the optimal voltage threshold is
0.995687\sigma. [In my previous recent e-mail I mistakenly wrote
0.9995687] Twice the reciprocal of that value is 2.00866, which rounds to
2.01. Note that where Gianni writes "2.01 times the quantization step in
2 bit TDM mode", the quantization step he is evidently referring to is the
3-bit quantizer step before discarding the lsb. So that's why you want
twice the reciprocal value for the effective 2-bit quantization step.
This morning I had a look again at the
data sheet describing the ALMA 3-bit, 4Gsps ADC.
Table 3 on p. 7 of this document (from Alain Baudry,
see attachment) shows the Gray code output of the ALMA sampler as a
function of input voltage. In my previous email, I neglected to take
cognizance of the fact that the encoding scheme for this output is a 3-bit
cyclic Gray code, rather than two's complement format. But the result is
equivalent: discarding the least-significant bit from the 3-bit output
yields the appropriate 2-bit Gray code. Here's a modified table:
3-bit 2-bit
Gray code discard lsb Gray code
+7 <- 100 -> 10 \
-> +3
+5 <- 101 -> 10 /
+3 <- 111 -> 11 \
-> +1
+1 <- 110 -> 11 /
-1 <- 010 -> 01 \
-> -1
-3 <- 011 -> 01 /
-5 <- 001 -> 00 \
-> -3
-7 <- 000 -> 00 /
Here I've corrected the earlier paragraph of mine:
"Regarding footnote 2 of Gianni's memo, I probably should re-think
this tomorrow, but note that when you discard the lsb from the 3-bit
2's complement levels, you effectively get double the voltage threshold
spacings. So the threshold spacings become 2*0.586019\sigma, and
the ratio of that to 0.995687\sigma is 1.17712. That is a voltage
ratio, and 20 Log_{10}(1.17712) = 1.41638 dB, whose rounded value is
equal to Gianni's 1.4 dB."
- Fred
--
ToddHunter - 2013-09-12